Counting multicurves on combinatorial surfaces and integrals of volumes of polytopes

  • Gaetan Borot (Humboldt University Berlin)
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Combinatorial surfaces can be seen as the limit of bordered hyperbolic surfaces with large boundaries, and we can study the problem of counting multicurves on them.

For a given combinatorial structure, multicurves correspond to integral points in a polytope defined in terms of ribbon graphs, and the asymptotic number of multicurves of very large length is controlled by the volume of those polytopes. This volume is a function on the moduli space of bordered surfaces, and motivated by questions from random geometry, one can ask for which s is this function $L^s$. The analog of this question can be posed in the hyperbolic case, and the answer is: $s \leq 2$. In the hyperbolic case, recent results of Arana-Herrera show $L^2$. In the combinatorial case, we show using methods from convex geometry and analysis of divergences from subgraphs that the answer is $s < s_{g,n}$ for an explicit $s_{g,n}$ depending on the genus $g$ and the number of boundaries $n$. This leads to various (analytic and arithmetic) questions around such enumeration problems, which can be compared to questions about Feynman integrals.

This is a joint work with Charbonnier, Delecroix, Giacchetto and Wheeler.

4/22/21 1/14/22

Leipzig seminar on Algebra, Algebraic Geometry and Algebraic Topology

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Katharina Matschke

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